Maximal asymptotic power and efficiency of two-sample tests based on generalized U-Statistics

In this article a systematic study is given of the asymptotic behavior of two-sample tests based on U-Statistics with arbitrary antisymmetric kernels ψ. Besides the investigation under the hypothesis and under fixed alternatives we determine the local power as a function of ψ as well as its maximizing value ψ<Subscript>opt</Subscript>. Moreover formulas for the asymptotic relative efficiency ARE(ψ<Subscript>2</Subscript>,ψ<Subscript>1</Subscript>) of the ψ<Subscript>2</Subscript>-test with respect to the ψ<Subscript>1</Subscript>-test are derived. It turns out that ψ<Subscript>opt</Subscript> also yields the most efficient test in the sense that ARE(ψ<Subscript>opt</Subscript>,ψ)≤1 for all (admissible) kernels ψ. Copyright Springer-Verlag 2004